Jordan-Holder theorem: Difference between revisions
(New page: {{semibasic fact}} ==Statement== Suppose <math>G</math> is a fact about::group of finite composition length. In other words, <math>G</math> has a fact about::composition series o...) |
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==Statement== | ==Statement== | ||
Suppose <math>G</math> is a [[ | Suppose <math>G</math> is a [[group of finite composition length]]. In other words, <math>G</math> has a [[composition series]] of finite length <math>l</math>: | ||
<math>\{ e \} = N_0 | <math>\{ e \} = N_0 \triangleleft N_1 \triangleleft N_2 \triangleleft \dots \triangleleft N_l = G</math> | ||
where each <math>N_{i-1}</math> is a [[proper normal subgroup]] of <math>N_i</math> and <math>N_i/N_{i-1}</math> is a [[simple group]]. Then, the following are true: | where each <math>N_{i-1}</math> is a [[proper normal subgroup]] of <math>N_i</math> and <math>N_i/N_{i-1}</math> is a [[simple group]]. Then, the following are true: | ||
Latest revision as of 00:59, 14 June 2024
This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic
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Statement
Suppose is a group of finite composition length. In other words, has a composition series of finite length :
where each is a proper normal subgroup of and is a simple group. Then, the following are true:
- Any composition series for has length .
- The list of composition factors is the same for any two composition series. In other words, if form one composition series and form another, then for any simple group , the number of for which is isomorphic to equals the number of for which is isomorphic to .
Related facts
- Jordan-Holder theorem for chief series: An analogous result, which states that any two chief series of a group have the same length and that the list of chief factors is the same.
- Finite composition length implies every subnormal series can be refined to a composition series
- Finite chief length implies every normal series can be refined to a chief series
- Jordan-Holder theorem for groups with operators
Some other related facts:
- Finite not implies composition factor-unique: There can exist finite groups for which there are different composition series with the composition factors occurring in different orders.
- Composition factor-unique not implies composition series-unique: Even if all composition series for a group have the same composition factors occurring in the same order, there may be more than one composition series.
- Finite not implies composition factor-permutable: There can exist finite groups for which not all possible orderings of the composition factors can be achieved using composition series.